Find the value separating the bottom 40% values from the top 60% values. Find the sample mean separating the bottom 40% sample means from the top 60% sample means.

Question

A population of values has a normal distribution with μ=113μ=113 and σ=89.3σ=89.3. You intend to draw a random sample of size n=198n=198. Please answer the following questions, and show your answers to 1 decimal place.

Answer 1

A population of values has a normal distribution with μ=113 and σ=89.3. You intend to draw a random sample of size n=198. Please answer the following questions, and show your answers to 1 decimal place.

Find the value separating the bottom 40% values from the top 60% values. Find the sample mean separating the bottom 40% of sample means from the top 60% of sample means.

Step-1

Given:

Population is normally distributed

Population mean μ=113 

Population standard deviation σ=89.3

Sample size n=198

Step-2

Compute z-scores at 40% (cumulative probability at 0.40 from left). This is the point that separates bottom 40% from top 60%.

z-score at 0.40 = -0.253

Compute sample Standard Error SE =  σ / sqrt(n) =89.3/SQRT(198) = 6.35

Step-3

Compute sample mean value from sampling distribution of sample means at 40% cumulative probability from left:

x = z * SE + μ = -0.253*6.35 + 113 = 111.39

Answer:

Sample mean separating the bottom 40% of sample means from the top 60% of sample means = 111.39 = 111.4